LMFDB - Embedded newform 272.10.a.f.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [272,10,Mod(1,272)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(272, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 10, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("272.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 272 = 2^{4} \cdot 17 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	272.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$140.089747437$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 1596x^{3} + 5754x^{2} + 488987x - 2711704 $ x^5 - 2x^4 - 1596x^3 + 5754x^2 + 488987x - 2711704
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3 $
Twist minimal:	no (minimal twist has level 17)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$33.6330$ of defining polynomial
Character	$\chi$	$=$	272.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+85.9747 q^{3} -1460.58 q^{5} +446.232 q^{7} -12291.3 q^{9} +O(q^{10})$	$q+85.9747 q^{3} -1460.58 q^{5} +446.232 q^{7} -12291.3 q^{9} -2565.20 q^{11} +70467.9 q^{13} -125573. q^{15} -83521.0 q^{17} +474111. q^{19} +38364.6 q^{21} -1.74586e6 q^{23} +180168. q^{25} -2.74899e6 q^{27} +501157. q^{29} +510635. q^{31} -220542. q^{33} -651757. q^{35} -3.74847e6 q^{37} +6.05845e6 q^{39} -3.04327e7 q^{41} +2.15222e7 q^{43} +1.79525e7 q^{45} -3.43306e6 q^{47} -4.01545e7 q^{49} -7.18069e6 q^{51} -2.87800e7 q^{53} +3.74668e6 q^{55} +4.07615e7 q^{57} +1.37313e8 q^{59} +1.21854e8 q^{61} -5.48479e6 q^{63} -1.02924e8 q^{65} +8.31665e7 q^{67} -1.50100e8 q^{69} -1.48492e8 q^{71} +1.45029e8 q^{73} +1.54899e7 q^{75} -1.14467e6 q^{77} +4.22172e8 q^{79} +5.58742e6 q^{81} +5.36072e7 q^{83} +1.21989e8 q^{85} +4.30868e7 q^{87} +9.26167e8 q^{89} +3.14450e7 q^{91} +4.39017e7 q^{93} -6.92477e8 q^{95} +1.44334e9 q^{97} +3.15298e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 236 q^{3} + 1480 q^{5} + 13202 q^{7} + 10981 q^{9}+O(q^{10}) $ 5 * q + 236 * q^3 + 1480 * q^5 + 13202 * q^7 + 10981 * q^9	$ 5 q + 236 q^{3} + 1480 q^{5} + 13202 q^{7} + 10981 q^{9} + 68036 q^{11} - 158862 q^{13} + 687324 q^{15} - 417605 q^{17} + 370992 q^{19} + 1783880 q^{21} - 1645870 q^{23} + 3270239 q^{25} + 2998268 q^{27} + 3668616 q^{29} + 7262362 q^{31} - 11334900 q^{33} + 26503988 q^{35} - 31420708 q^{37} + 42449884 q^{39} - 7996938 q^{41} + 56908268 q^{43} + 12799536 q^{45} + 16903336 q^{47} - 11784059 q^{49} - 19710956 q^{51} - 83362982 q^{53} - 6363364 q^{55} + 136615904 q^{57} + 37946604 q^{59} - 77685452 q^{61} + 191945278 q^{63} - 40321288 q^{65} + 304503600 q^{67} - 333409272 q^{69} + 476602922 q^{71} - 289980486 q^{73} + 153685772 q^{75} - 143385648 q^{77} + 828240610 q^{79} + 891328609 q^{81} - 194681148 q^{83} - 123611080 q^{85} - 158149884 q^{87} + 376848106 q^{89} - 194543664 q^{91} + 3494835920 q^{93} - 1498679864 q^{95} + 692035246 q^{97} - 2027106408 q^{99}+O(q^{100}) $ 5 * q + 236 * q^3 + 1480 * q^5 + 13202 * q^7 + 10981 * q^9 + 68036 * q^11 - 158862 * q^13 + 687324 * q^15 - 417605 * q^17 + 370992 * q^19 + 1783880 * q^21 - 1645870 * q^23 + 3270239 * q^25 + 2998268 * q^27 + 3668616 * q^29 + 7262362 * q^31 - 11334900 * q^33 + 26503988 * q^35 - 31420708 * q^37 + 42449884 * q^39 - 7996938 * q^41 + 56908268 * q^43 + 12799536 * q^45 + 16903336 * q^47 - 11784059 * q^49 - 19710956 * q^51 - 83362982 * q^53 - 6363364 * q^55 + 136615904 * q^57 + 37946604 * q^59 - 77685452 * q^61 + 191945278 * q^63 - 40321288 * q^65 + 304503600 * q^67 - 333409272 * q^69 + 476602922 * q^71 - 289980486 * q^73 + 153685772 * q^75 - 143385648 * q^77 + 828240610 * q^79 + 891328609 * q^81 - 194681148 * q^83 - 123611080 * q^85 - 158149884 * q^87 + 376848106 * q^89 - 194543664 * q^91 + 3494835920 * q^93 - 1498679864 * q^95 + 692035246 * q^97 - 2027106408 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	85.9747	0.612809	0.306404	−	0.951901i	$-0.400874\pi$
$3$	85.9747	0.612809	0.306404	+	0.951901i	$0.400874\pi$
$4$	0	0
$5$	−1460.58	−1.04511	−0.522553	−	0.852607i	$-0.675020\pi$
$5$	−1460.58	−1.04511	−0.522553	+	0.852607i	$0.675020\pi$
$6$	0	0
$7$	446.232	0.0702456	0.0351228	−	0.999383i	$-0.488818\pi$
$7$	446.232	0.0702456	0.0351228	+	0.999383i	$0.488818\pi$
$8$	0	0
$9$	−12291.3	−0.624465
$10$	0	0
$11$	−2565.20	−0.0528268	−0.0264134	−	0.999651i	$-0.508409\pi$
$11$	−2565.20	−0.0528268	−0.0264134	+	0.999651i	$0.508409\pi$
$12$	0	0
$13$	70467.9	0.684299	0.342150	−	0.939645i	$-0.388845\pi$
$13$	70467.9	0.684299	0.342150	+	0.939645i	$0.388845\pi$
$14$	0	0
$15$	−125573.	−0.640450
$16$	0	0
$17$	−83521.0	−0.242536
$17$	−83521.0	−0.242536
$18$	0	0
$19$	474111.	0.834620	0.417310	−	0.908764i	$-0.362973\pi$
$19$	474111.	0.834620	0.417310	+	0.908764i	$0.362973\pi$
$20$	0	0
$21$	38364.6	0.0430471
$22$	0	0
$23$	−1.74586e6	−1.30087	−0.650437	−	0.759561i	$-0.725414\pi$
$23$	−1.74586e6	−1.30087	−0.650437	+	0.759561i	$0.725414\pi$
$24$	0	0
$25$	180168.	0.0922461
$26$	0	0
$27$	−2.74899e6	−0.995487
$28$	0	0
$29$	501157.	0.131578	0.0657889	−	0.997834i	$-0.479044\pi$
$29$	501157.	0.131578	0.0657889	+	0.997834i	$0.479044\pi$
$30$	0	0
$31$	510635.	0.0993077	0.0496538	−	0.998766i	$-0.484188\pi$
$31$	510635.	0.0993077	0.0496538	+	0.998766i	$0.484188\pi$
$32$	0	0
$33$	−220542.	−0.0323727
$34$	0	0
$35$	−651757.	−0.0734141
$36$	0	0
$37$	−3.74847e6	−0.328811	−0.164406	−	0.986393i	$-0.552571\pi$
$37$	−3.74847e6	−0.328811	−0.164406	+	0.986393i	$0.552571\pi$
$38$	0	0
$39$	6.05845e6	0.419345
$40$	0	0
$41$	−3.04327e7	−1.68195	−0.840975	−	0.541074i	$-0.818018\pi$
$41$	−3.04327e7	−1.68195	−0.840975	+	0.541074i	$0.818018\pi$
$42$	0	0
$43$	2.15222e7	0.960017	0.480008	−	0.877264i	$-0.340634\pi$
$43$	2.15222e7	0.960017	0.480008	+	0.877264i	$0.340634\pi$
$44$	0	0
$45$	1.79525e7	0.652632
$46$	0	0
$47$	−3.43306e6	−0.102622	−0.0513111	−	0.998683i	$-0.516340\pi$
$47$	−3.43306e6	−0.102622	−0.0513111	+	0.998683i	$0.516340\pi$
$48$	0	0
$49$	−4.01545e7	−0.995066
$50$	0	0
$51$	−7.18069e6	−0.148628
$52$	0	0
$53$	−2.87800e7	−0.501013	−0.250506	−	0.968115i	$-0.580597\pi$
$53$	−2.87800e7	−0.501013	−0.250506	+	0.968115i	$0.580597\pi$
$54$	0	0
$55$	3.74668e6	0.0552096
$56$	0	0
$57$	4.07615e7	0.511462
$58$	0	0
$59$	1.37313e8	1.47529	0.737643	−	0.675191i	$-0.235939\pi$
$59$	1.37313e8	1.47529	0.737643	+	0.675191i	$0.235939\pi$
$60$	0	0
$61$	1.21854e8	1.12682	0.563410	−	0.826177i	$-0.309489\pi$
$61$	1.21854e8	1.12682	0.563410	+	0.826177i	$0.309489\pi$
$62$	0	0
$63$	−5.48479e6	−0.0438659
$64$	0	0
$65$	−1.02924e8	−0.715165
$66$	0	0
$67$	8.31665e7	0.504210	0.252105	−	0.967700i	$-0.418877\pi$
$67$	8.31665e7	0.504210	0.252105	+	0.967700i	$0.418877\pi$
$68$	0	0
$69$	−1.50100e8	−0.797187
$70$	0	0
$71$	−1.48492e8	−0.693490	−0.346745	−	0.937960i	$-0.612713\pi$
$71$	−1.48492e8	−0.693490	−0.346745	+	0.937960i	$0.612713\pi$
$72$	0	0
$73$	1.45029e8	0.597726	0.298863	−	0.954296i	$-0.403393\pi$
$73$	1.45029e8	0.597726	0.298863	+	0.954296i	$0.403393\pi$
$74$	0	0
$75$	1.54899e7	0.0565292
$76$	0	0
$77$	−1.14467e6	−0.00371085
$78$	0	0
$79$	4.22172e8	1.21946	0.609729	−	0.792610i	$-0.291278\pi$
$79$	4.22172e8	1.21946	0.609729	+	0.792610i	$0.291278\pi$
$80$	0	0
$81$	5.58742e6	0.0144221
$82$	0	0
$83$	5.36072e7	0.123986	0.0619929	−	0.998077i	$-0.480254\pi$
$83$	5.36072e7	0.123986	0.0619929	+	0.998077i	$0.480254\pi$
$84$	0	0
$85$	1.21989e8	0.253475
$86$	0	0
$87$	4.30868e7	0.0806321
$88$	0	0
$89$	9.26167e8	1.56471	0.782355	−	0.622832i	$-0.214018\pi$
$89$	9.26167e8	1.56471	0.782355	+	0.622832i	$0.214018\pi$
$90$	0	0
$91$	3.14450e7	0.0480690
$92$	0	0
$93$	4.39017e7	0.0608566
$94$	0	0
$95$	−6.92477e8	−0.872266
$96$	0	0
$97$	1.44334e9	1.65538	0.827688	−	0.561189i	$-0.189656\pi$
$97$	1.44334e9	1.65538	0.827688	+	0.561189i	$0.189656\pi$
$98$	0	0
$99$	3.15298e7	0.0329885
$100$	0	0
$101$	1.73497e9	1.65900	0.829498	−	0.558510i	$-0.188627\pi$
$101$	1.73497e9	1.65900	0.829498	+	0.558510i	$0.188627\pi$
$102$	0	0
$103$	1.15796e9	1.01374	0.506871	−	0.862022i	$-0.330802\pi$
$103$	1.15796e9	1.01374	0.506871	+	0.862022i	$0.330802\pi$
$104$	0	0
$105$	−5.60346e7	−0.0449888
$106$	0	0
$107$	−6.18767e8	−0.456352	−0.228176	−	0.973620i	$-0.573276\pi$
$107$	−6.18767e8	−0.456352	−0.228176	+	0.973620i	$0.573276\pi$
$108$	0	0
$109$	−1.81586e9	−1.23215	−0.616074	−	0.787688i	$-0.711278\pi$
$109$	−1.81586e9	−1.23215	−0.616074	+	0.787688i	$0.711278\pi$
$110$	0	0
$111$	−3.22274e8	−0.201498
$112$	0	0
$113$	−2.89208e9	−1.66862	−0.834310	−	0.551296i	$-0.814134\pi$
$113$	−2.89208e9	−1.66862	−0.834310	+	0.551296i	$0.814134\pi$
$114$	0	0
$115$	2.54997e9	1.35955
$116$	0	0
$117$	−8.66145e8	−0.427321
$118$	0	0
$119$	−3.72697e7	−0.0170371
$120$	0	0
$121$	−2.35137e9	−0.997209
$122$	0	0
$123$	−2.61644e9	−1.03071
$124$	0	0
$125$	2.58954e9	0.948699
$126$	0	0
$127$	−1.38692e9	−0.473081	−0.236540	−	0.971622i	$-0.576014\pi$
$127$	−1.38692e9	−0.473081	−0.236540	+	0.971622i	$0.576014\pi$
$128$	0	0
$129$	1.85037e9	0.588307
$130$	0	0
$131$	−5.87855e9	−1.74401	−0.872006	−	0.489496i	$-0.837181\pi$
$131$	−5.87855e9	−1.74401	−0.872006	+	0.489496i	$0.837181\pi$
$132$	0	0
$133$	2.11563e8	0.0586284
$134$	0	0
$135$	4.01511e9	1.04039
$136$	0	0
$137$	2.25574e9	0.547075	0.273538	−	0.961861i	$-0.411806\pi$
$137$	2.25574e9	0.547075	0.273538	+	0.961861i	$0.411806\pi$
$138$	0	0
$139$	2.06541e9	0.469289	0.234645	−	0.972081i	$-0.424607\pi$
$139$	2.06541e9	0.469289	0.234645	+	0.972081i	$0.424607\pi$
$140$	0	0
$141$	−2.95156e8	−0.0628878
$142$	0	0
$143$	−1.80764e8	−0.0361493
$144$	0	0
$145$	−7.31980e8	−0.137513
$146$	0	0
$147$	−3.45227e9	−0.609785
$148$	0	0
$149$	−3.00126e9	−0.498845	−0.249422	−	0.968395i	$-0.580241\pi$
$149$	−3.00126e9	−0.498845	−0.249422	+	0.968395i	$0.580241\pi$
$150$	0	0
$151$	1.05501e10	1.65143	0.825714	−	0.564089i	$-0.190773\pi$
$151$	1.05501e10	1.65143	0.825714	+	0.564089i	$0.190773\pi$
$152$	0	0
$153$	1.02659e9	0.151455
$154$	0	0
$155$	−7.45823e8	−0.103787
$156$	0	0
$157$	8.51695e9	1.11876	0.559378	−	0.828912i	$-0.311040\pi$
$157$	8.51695e9	1.11876	0.559378	+	0.828912i	$0.311040\pi$
$158$	0	0
$159$	−2.47435e9	−0.307025
$160$	0	0
$161$	−7.79059e8	−0.0913806
$162$	0	0
$163$	6.90475e9	0.766133	0.383066	−	0.923721i	$-0.374868\pi$
$163$	6.90475e9	0.766133	0.383066	+	0.923721i	$0.374868\pi$
$164$	0	0
$165$	3.22120e8	0.0338329
$166$	0	0
$167$	8.18088e9	0.813909	0.406954	−	0.913448i	$-0.366591\pi$
$167$	8.18088e9	0.813909	0.406954	+	0.913448i	$0.366591\pi$
$168$	0	0
$169$	−5.63878e9	−0.531735
$170$	0	0
$171$	−5.82746e9	−0.521191
$172$	0	0
$173$	1.96517e9	0.166799	0.0833995	−	0.996516i	$-0.473422\pi$
$173$	1.96517e9	0.166799	0.0833995	+	0.996516i	$0.473422\pi$
$174$	0	0
$175$	8.03968e7	0.00647988
$176$	0	0
$177$	1.18054e10	0.904069
$178$	0	0
$179$	2.51083e10	1.82801	0.914006	−	0.405701i	$-0.132972\pi$
$179$	2.51083e10	1.82801	0.914006	+	0.405701i	$0.132972\pi$
$180$	0	0
$181$	−3.03563e9	−0.210230	−0.105115	−	0.994460i	$-0.533521\pi$
$181$	−3.03563e9	−0.210230	−0.105115	+	0.994460i	$0.533521\pi$
$182$	0	0
$183$	1.04763e10	0.690526
$184$	0	0
$185$	5.47494e9	0.343642
$186$	0	0
$187$	2.14248e8	0.0128124
$188$	0	0
$189$	−1.22668e9	−0.0699286
$190$	0	0
$191$	3.15580e10	1.71577	0.857885	−	0.513841i	$-0.171778\pi$
$191$	3.15580e10	1.71577	0.857885	+	0.513841i	$0.171778\pi$
$192$	0	0
$193$	2.29526e9	0.119076	0.0595379	−	0.998226i	$-0.481037\pi$
$193$	2.29526e9	0.119076	0.0595379	+	0.998226i	$0.481037\pi$
$194$	0	0
$195$	−8.84886e9	−0.438260
$196$	0	0
$197$	2.65359e10	1.25527	0.627633	−	0.778509i	$-0.284024\pi$
$197$	2.65359e10	1.25527	0.627633	+	0.778509i	$0.284024\pi$
$198$	0	0
$199$	1.45778e10	0.658949	0.329475	−	0.944164i	$-0.393128\pi$
$199$	1.45778e10	0.658949	0.329475	+	0.944164i	$0.393128\pi$
$200$	0	0
$201$	7.15021e9	0.308985
$202$	0	0
$203$	2.23632e8	0.00924276
$204$	0	0
$205$	4.44494e10	1.75782
$206$	0	0
$207$	2.14590e10	0.812350
$208$	0	0
$209$	−1.21619e9	−0.0440903
$210$	0	0
$211$	−4.31184e10	−1.49759	−0.748793	−	0.662804i	$-0.769366\pi$
$211$	−4.31184e10	−1.49759	−0.748793	+	0.662804i	$0.769366\pi$
$212$	0	0
$213$	−1.27665e10	−0.424977
$214$	0	0
$215$	−3.14349e10	−1.00332
$216$	0	0
$217$	2.27861e8	0.00697593
$218$	0	0
$219$	1.24688e10	0.366292
$220$	0	0
$221$	−5.88555e9	−0.165967
$222$	0	0
$223$	3.81093e10	1.03195	0.515975	−	0.856604i	$-0.327430\pi$
$223$	3.81093e10	1.03195	0.515975	+	0.856604i	$0.327430\pi$
$224$	0	0
$225$	−2.21451e9	−0.0576045
$226$	0	0
$227$	2.78123e10	0.695216	0.347608	−	0.937640i	$-0.386994\pi$
$227$	2.78123e10	0.695216	0.347608	+	0.937640i	$0.386994\pi$
$228$	0	0
$229$	3.54756e10	0.852452	0.426226	−	0.904617i	$-0.359843\pi$
$229$	3.54756e10	0.852452	0.426226	+	0.904617i	$0.359843\pi$
$230$	0	0
$231$	−9.84130e7	−0.00227404
$232$	0	0
$233$	4.62674e10	1.02843	0.514213	−	0.857662i	$-0.328084\pi$
$233$	4.62674e10	1.02843	0.514213	+	0.857662i	$0.328084\pi$
$234$	0	0
$235$	5.01426e9	0.107251
$236$	0	0
$237$	3.62961e10	0.747295
$238$	0	0
$239$	−8.33772e10	−1.65294	−0.826469	−	0.562982i	$-0.809654\pi$
$239$	−8.33772e10	−1.65294	−0.826469	+	0.562982i	$0.809654\pi$
$240$	0	0
$241$	−7.31412e9	−0.139664	−0.0698321	−	0.997559i	$-0.522246\pi$
$241$	−7.31412e9	−0.139664	−0.0698321	+	0.997559i	$0.522246\pi$
$242$	0	0
$243$	5.45887e10	1.00432
$244$	0	0
$245$	5.86488e10	1.03995
$246$	0	0
$247$	3.34096e10	0.571130
$248$	0	0
$249$	4.60886e9	0.0759795
$250$	0	0
$251$	−9.50519e10	−1.51157	−0.755786	−	0.654818i	$-0.772745\pi$
$251$	−9.50519e10	−1.51157	−0.755786	+	0.654818i	$0.772745\pi$
$252$	0	0
$253$	4.47849e9	0.0687210
$254$	0	0
$255$	1.04880e10	0.155332
$256$	0	0
$257$	−7.20648e10	−1.03044	−0.515222	−	0.857057i	$-0.672290\pi$
$257$	−7.20648e10	−1.03044	−0.515222	+	0.857057i	$0.672290\pi$
$258$	0	0
$259$	−1.67269e9	−0.0230975
$260$	0	0
$261$	−6.15989e9	−0.0821658
$262$	0	0
$263$	1.71832e10	0.221464	0.110732	−	0.993850i	$-0.464680\pi$
$263$	1.71832e10	0.221464	0.110732	+	0.993850i	$0.464680\pi$
$264$	0	0
$265$	4.20354e10	0.523612
$266$	0	0
$267$	7.96269e10	0.958869
$268$	0	0
$269$	1.10544e10	0.128721	0.0643605	−	0.997927i	$-0.479499\pi$
$269$	1.10544e10	0.128721	0.0643605	+	0.997927i	$0.479499\pi$
$270$	0	0
$271$	1.41142e11	1.58963	0.794813	−	0.606855i	$-0.207569\pi$
$271$	1.41142e11	1.58963	0.794813	+	0.606855i	$0.207569\pi$
$272$	0	0
$273$	2.70347e9	0.0294571
$274$	0	0
$275$	−4.62168e8	−0.00487307
$276$	0	0
$277$	−1.12712e11	−1.15030	−0.575151	−	0.818047i	$-0.695057\pi$
$277$	−1.12712e11	−1.15030	−0.575151	+	0.818047i	$0.695057\pi$
$278$	0	0
$279$	−6.27639e9	−0.0620142
$280$	0	0
$281$	4.86687e10	0.465662	0.232831	−	0.972517i	$-0.425201\pi$
$281$	4.86687e10	0.465662	0.232831	+	0.972517i	$0.425201\pi$
$282$	0	0
$283$	1.21207e11	1.12328	0.561640	−	0.827382i	$-0.310171\pi$
$283$	1.21207e11	1.12328	0.561640	+	0.827382i	$0.310171\pi$
$284$	0	0
$285$	−5.95355e10	−0.534532
$286$	0	0
$287$	−1.35800e10	−0.118150
$288$	0	0
$289$	6.97576e9	0.0588235
$290$	0	0
$291$	1.24091e11	1.01443
$292$	0	0
$293$	9.65141e10	0.765044	0.382522	−	0.923946i	$-0.375056\pi$
$293$	9.65141e10	0.765044	0.382522	+	0.923946i	$0.375056\pi$
$294$	0	0
$295$	−2.00556e11	−1.54183
$296$	0	0
$297$	7.05170e9	0.0525884
$298$	0	0
$299$	−1.23027e11	−0.890187
$300$	0	0
$301$	9.60389e9	0.0674370
$302$	0	0
$303$	1.49163e11	1.01665
$304$	0	0
$305$	−1.77977e11	−1.17765
$306$	0	0
$307$	1.31572e11	0.845360	0.422680	−	0.906279i	$-0.361089\pi$
$307$	1.31572e11	0.845360	0.422680	+	0.906279i	$0.361089\pi$
$308$	0	0
$309$	9.95556e10	0.621230
$310$	0	0
$311$	−3.19848e9	−0.0193875	−0.00969376	−	0.999953i	$-0.503086\pi$
$311$	−3.19848e9	−0.0193875	−0.00969376	+	0.999953i	$0.503086\pi$
$312$	0	0
$313$	−1.59936e11	−0.941882	−0.470941	−	0.882165i	$-0.656086\pi$
$313$	−1.59936e11	−0.941882	−0.470941	+	0.882165i	$0.656086\pi$
$314$	0	0
$315$	8.01097e9	0.0458445
$316$	0	0
$317$	6.77437e10	0.376792	0.188396	−	0.982093i	$-0.439671\pi$
$317$	6.77437e10	0.376792	0.188396	+	0.982093i	$0.439671\pi$
$318$	0	0
$319$	−1.28557e9	−0.00695083
$320$	0	0
$321$	−5.31983e10	−0.279657
$322$	0	0
$323$	−3.95982e10	−0.202425
$324$	0	0
$325$	1.26961e10	0.0631239
$326$	0	0
$327$	−1.56118e11	−0.755071
$328$	0	0
$329$	−1.53194e9	−0.00720876
$330$	0	0
$331$	−1.23890e11	−0.567295	−0.283647	−	0.958929i	$-0.591544\pi$
$331$	−1.23890e11	−0.567295	−0.283647	+	0.958929i	$0.591544\pi$
$332$	0	0
$333$	4.60738e10	0.205331
$334$	0	0
$335$	−1.21471e11	−0.526953
$336$	0	0
$337$	−1.66091e11	−0.701476	−0.350738	−	0.936474i	$-0.614069\pi$
$337$	−1.66091e11	−0.701476	−0.350738	+	0.936474i	$0.614069\pi$
$338$	0	0
$339$	−2.48646e11	−1.02254
$340$	0	0
$341$	−1.30988e9	−0.00524611
$342$	0	0
$343$	−3.59253e10	−0.140145
$344$	0	0
$345$	2.19233e11	0.833144
$346$	0	0
$347$	3.59340e11	1.33052	0.665262	−	0.746610i	$-0.268320\pi$
$347$	3.59340e11	1.33052	0.665262	+	0.746610i	$0.268320\pi$
$348$	0	0
$349$	4.55698e11	1.64423	0.822116	−	0.569321i	$-0.192794\pi$
$349$	4.55698e11	1.64423	0.822116	+	0.569321i	$0.192794\pi$
$350$	0	0
$351$	−1.93715e11	−0.681211
$352$	0	0
$353$	−2.35070e11	−0.805772	−0.402886	−	0.915250i	$-0.631993\pi$
$353$	−2.35070e11	−0.805772	−0.402886	+	0.915250i	$0.631993\pi$
$354$	0	0
$355$	2.16884e11	0.724770
$356$	0	0
$357$	−3.20425e9	−0.0104405
$358$	0	0
$359$	−8.46696e10	−0.269031	−0.134516	−	0.990911i	$-0.542948\pi$
$359$	−8.46696e10	−0.269031	−0.134516	+	0.990911i	$0.542948\pi$
$360$	0	0
$361$	−9.79067e10	−0.303410
$362$	0	0
$363$	−2.02158e11	−0.611099
$364$	0	0
$365$	−2.11827e11	−0.624687
$366$	0	0
$367$	−3.10781e10	−0.0894246	−0.0447123	−	0.999000i	$-0.514237\pi$
$367$	−3.10781e10	−0.0894246	−0.0447123	+	0.999000i	$0.514237\pi$
$368$	0	0
$369$	3.74059e11	1.05032
$370$	0	0
$371$	−1.28425e10	−0.0351940
$372$	0	0
$373$	−6.64364e11	−1.77712	−0.888559	−	0.458763i	$-0.848293\pi$
$373$	−6.64364e11	−1.77712	−0.888559	+	0.458763i	$0.848293\pi$
$374$	0	0
$375$	2.22635e11	0.581371
$376$	0	0
$377$	3.53155e10	0.0900386
$378$	0	0
$379$	1.05029e11	0.261475	0.130738	−	0.991417i	$-0.458265\pi$
$379$	1.05029e11	0.261475	0.130738	+	0.991417i	$0.458265\pi$
$380$	0	0
$381$	−1.19240e11	−0.289908
$382$	0	0
$383$	−6.90072e11	−1.63870	−0.819351	−	0.573293i	$-0.805666\pi$
$383$	−6.90072e11	−1.63870	−0.819351	+	0.573293i	$0.805666\pi$
$384$	0	0
$385$	1.67189e9	0.00387823
$386$	0	0
$387$	−2.64537e11	−0.599497
$388$	0	0
$389$	1.43485e11	0.317712	0.158856	−	0.987302i	$-0.449219\pi$
$389$	1.43485e11	0.317712	0.158856	+	0.987302i	$0.449219\pi$
$390$	0	0
$391$	1.45816e11	0.315508
$392$	0	0
$393$	−5.05407e11	−1.06875
$394$	0	0
$395$	−6.16615e11	−1.27446
$396$	0	0
$397$	−4.59121e11	−0.927620	−0.463810	−	0.885935i	$-0.653518\pi$
$397$	−4.59121e11	−0.927620	−0.463810	+	0.885935i	$0.653518\pi$
$398$	0	0
$399$	1.81891e10	0.0359280
$400$	0	0
$401$	−1.18477e11	−0.228815	−0.114408	−	0.993434i	$-0.536497\pi$
$401$	−1.18477e11	−0.228815	−0.114408	+	0.993434i	$0.536497\pi$
$402$	0	0
$403$	3.59834e10	0.0679562
$404$	0	0
$405$	−8.16088e9	−0.0150726
$406$	0	0
$407$	9.61558e9	0.0173700
$408$	0	0
$409$	3.44816e11	0.609301	0.304651	−	0.952464i	$-0.401460\pi$
$409$	3.44816e11	0.609301	0.304651	+	0.952464i	$0.401460\pi$
$410$	0	0
$411$	1.93937e11	0.335253
$412$	0	0
$413$	6.12732e10	0.103632
$414$	0	0
$415$	−7.82976e10	−0.129578
$416$	0	0
$417$	1.77573e11	0.287585
$418$	0	0
$419$	−1.06432e12	−1.68697	−0.843487	−	0.537150i	$-0.819501\pi$
$419$	−1.06432e12	−1.68697	−0.843487	+	0.537150i	$0.819501\pi$
$420$	0	0
$421$	8.40482e11	1.30394	0.651972	−	0.758243i	$-0.273942\pi$
$421$	8.40482e11	1.30394	0.651972	+	0.758243i	$0.273942\pi$
$422$	0	0
$423$	4.21970e10	0.0640840
$424$	0	0
$425$	−1.50478e10	−0.0223730
$426$	0	0
$427$	5.43750e10	0.0791542
$428$	0	0
$429$	−1.55411e10	−0.0221526
$430$	0	0
$431$	7.86921e11	1.09846	0.549229	−	0.835672i	$-0.314922\pi$
$431$	7.86921e11	1.09846	0.549229	+	0.835672i	$0.314922\pi$
$432$	0	0
$433$	−7.27832e11	−0.995029	−0.497514	−	0.867456i	$-0.665754\pi$
$433$	−7.27832e11	−0.995029	−0.497514	+	0.867456i	$0.665754\pi$
$434$	0	0
$435$	−6.29317e10	−0.0842690
$436$	0	0
$437$	−8.27733e11	−1.08573
$438$	0	0
$439$	5.63619e11	0.724261	0.362130	−	0.932127i	$-0.382050\pi$
$439$	5.63619e11	0.724261	0.362130	+	0.932127i	$0.382050\pi$
$440$	0	0
$441$	4.93553e11	0.621384
$442$	0	0
$443$	−2.48083e11	−0.306041	−0.153021	−	0.988223i	$-0.548900\pi$
$443$	−2.48083e11	−0.306041	−0.153021	+	0.988223i	$0.548900\pi$
$444$	0	0
$445$	−1.35274e12	−1.63529
$446$	0	0
$447$	−2.58033e11	−0.305697
$448$	0	0
$449$	−1.36225e12	−1.58178	−0.790891	−	0.611957i	$-0.790383\pi$
$449$	−1.36225e12	−1.58178	−0.790891	+	0.611957i	$0.790383\pi$
$450$	0	0
$451$	7.80660e10	0.0888520
$452$	0	0
$453$	9.07040e11	1.01201
$454$	0	0
$455$	−4.59279e10	−0.0502372
$456$	0	0
$457$	9.81977e11	1.05312	0.526560	−	0.850138i	$-0.323481\pi$
$457$	9.81977e11	1.05312	0.526560	+	0.850138i	$0.323481\pi$
$458$	0	0
$459$	2.29598e11	0.241441
$460$	0	0
$461$	7.36336e11	0.759314	0.379657	−	0.925127i	$-0.376042\pi$
$461$	7.36336e11	0.759314	0.379657	+	0.925127i	$0.376042\pi$
$462$	0	0
$463$	−1.79548e11	−0.181579	−0.0907894	−	0.995870i	$-0.528939\pi$
$463$	−1.79548e11	−0.181579	−0.0907894	+	0.995870i	$0.528939\pi$
$464$	0	0
$465$	−6.41219e10	−0.0636016
$466$	0	0
$467$	−1.13676e11	−0.110596	−0.0552982	−	0.998470i	$-0.517611\pi$
$467$	−1.13676e11	−0.110596	−0.0552982	+	0.998470i	$0.517611\pi$
$468$	0	0
$469$	3.71115e10	0.0354186
$470$	0	0
$471$	7.32242e11	0.685584
$472$	0	0
$473$	−5.52088e10	−0.0507146
$474$	0	0
$475$	8.54197e10	0.0769904
$476$	0	0
$477$	3.53745e11	0.312865
$478$	0	0
$479$	5.35347e11	0.464649	0.232325	−	0.972638i	$-0.425367\pi$
$479$	5.35347e11	0.464649	0.232325	+	0.972638i	$0.425367\pi$
$480$	0	0
$481$	−2.64147e11	−0.225005
$482$	0	0
$483$	−6.69794e10	−0.0559989
$484$	0	0
$485$	−2.10812e12	−1.73004
$486$	0	0
$487$	−9.30784e11	−0.749840	−0.374920	−	0.927057i	$-0.622330\pi$
$487$	−9.30784e11	−0.749840	−0.374920	+	0.927057i	$0.622330\pi$
$488$	0	0
$489$	5.93634e11	0.469493
$490$	0	0
$491$	−1.40150e12	−1.08824	−0.544122	−	0.839006i	$-0.683137\pi$
$491$	−1.40150e12	−1.08824	−0.544122	+	0.839006i	$0.683137\pi$
$492$	0	0
$493$	−4.18571e10	−0.0319123
$494$	0	0
$495$	−4.60517e10	−0.0344765
$496$	0	0
$497$	−6.62617e10	−0.0487146
$498$	0	0
$499$	−9.65319e11	−0.696977	−0.348488	−	0.937313i	$-0.613305\pi$
$499$	−9.65319e11	−0.696977	−0.348488	+	0.937313i	$0.613305\pi$
$500$	0	0
$501$	7.03348e11	0.498771
$502$	0	0
$503$	1.35251e12	0.942076	0.471038	−	0.882113i	$-0.343879\pi$
$503$	1.35251e12	0.942076	0.471038	+	0.882113i	$0.343879\pi$
$504$	0	0
$505$	−2.53406e12	−1.73383
$506$	0	0
$507$	−4.84792e11	−0.325852
$508$	0	0
$509$	−1.24943e12	−0.825053	−0.412527	−	0.910946i	$-0.635354\pi$
$509$	−1.24943e12	−0.825053	−0.412527	+	0.910946i	$0.635354\pi$
$510$	0	0
$511$	6.47166e10	0.0419876
$512$	0	0
$513$	−1.30332e12	−0.830853
$514$	0	0
$515$	−1.69130e12	−1.05947
$516$	0	0
$517$	8.80649e9	0.00542120
$518$	0	0
$519$	1.68955e11	0.102216
$520$	0	0
$521$	2.83113e11	0.168341	0.0841706	−	0.996451i	$-0.473176\pi$
$521$	2.83113e11	0.168341	0.0841706	+	0.996451i	$0.473176\pi$
$522$	0	0
$523$	−6.03608e11	−0.352774	−0.176387	−	0.984321i	$-0.556441\pi$
$523$	−6.03608e11	−0.352774	−0.176387	+	0.984321i	$0.556441\pi$
$524$	0	0
$525$	6.91209e9	0.00397093
$526$	0	0
$527$	−4.26487e10	−0.0240857
$528$	0	0
$529$	1.24689e12	0.692271
$530$	0	0
$531$	−1.68776e12	−0.921265
$532$	0	0
$533$	−2.14453e12	−1.15096
$534$	0	0
$535$	9.03758e11	0.476936
$536$	0	0
$537$	2.15868e12	1.12022
$538$	0	0
$539$	1.03004e11	0.0525661
$540$	0	0
$541$	−7.92682e11	−0.397843	−0.198921	−	0.980015i	$-0.563744\pi$
$541$	−7.92682e11	−0.397843	−0.198921	+	0.980015i	$0.563744\pi$
$542$	0	0
$543$	−2.60987e11	−0.128831
$544$	0	0
$545$	2.65221e12	1.28773
$546$	0	0
$547$	−5.81013e10	−0.0277487	−0.0138744	−	0.999904i	$-0.504416\pi$
$547$	−5.81013e10	−0.0277487	−0.0138744	+	0.999904i	$0.504416\pi$
$548$	0	0
$549$	−1.49775e12	−0.703660
$550$	0	0
$551$	2.37604e11	0.109817
$552$	0	0
$553$	1.88386e11	0.0856616
$554$	0	0
$555$	4.70707e11	0.210587
$556$	0	0
$557$	−8.04975e11	−0.354351	−0.177176	−	0.984179i	$-0.556696\pi$
$557$	−8.04975e11	−0.354351	−0.177176	+	0.984179i	$0.556696\pi$
$558$	0	0
$559$	1.51662e12	0.656939
$560$	0	0
$561$	1.84199e10	0.00785154
$562$	0	0
$563$	3.22746e12	1.35386	0.676929	−	0.736048i	$-0.263310\pi$
$563$	3.22746e12	1.35386	0.676929	+	0.736048i	$0.263310\pi$
$564$	0	0
$565$	4.22411e12	1.74388
$566$	0	0
$567$	2.49329e9	0.00101309
$568$	0	0
$569$	−2.00792e12	−0.803048	−0.401524	−	0.915848i	$-0.631519\pi$
$569$	−2.00792e12	−0.803048	−0.401524	+	0.915848i	$0.631519\pi$
$570$	0	0
$571$	−3.16004e11	−0.124403	−0.0622013	−	0.998064i	$-0.519812\pi$
$571$	−3.16004e11	−0.124403	−0.0622013	+	0.998064i	$0.519812\pi$
$572$	0	0
$573$	2.71319e12	1.05144
$574$	0	0
$575$	−3.14549e11	−0.120001
$576$	0	0
$577$	−1.81033e12	−0.679932	−0.339966	−	0.940438i	$-0.610416\pi$
$577$	−1.81033e12	−0.679932	−0.339966	+	0.940438i	$0.610416\pi$
$578$	0	0
$579$	1.97334e11	0.0729707
$580$	0	0
$581$	2.39212e10	0.00870945
$582$	0	0
$583$	7.38264e10	0.0264669
$584$	0	0
$585$	1.26507e12	0.446596
$586$	0	0
$587$	−2.94820e12	−1.02491	−0.512455	−	0.858714i	$-0.671264\pi$
$587$	−2.94820e12	−1.02491	−0.512455	+	0.858714i	$0.671264\pi$
$588$	0	0
$589$	2.42098e11	0.0828842
$590$	0	0
$591$	2.28142e12	0.769238
$592$	0	0
$593$	4.46478e12	1.48270	0.741350	−	0.671118i	$-0.234186\pi$
$593$	4.46478e12	1.48270	0.741350	+	0.671118i	$0.234186\pi$
$594$	0	0
$595$	5.44354e10	0.0178055
$596$	0	0
$597$	1.25332e12	0.403810
$598$	0	0
$599$	−3.15817e12	−1.00234	−0.501170	−	0.865349i	$-0.667097\pi$
$599$	−3.15817e12	−1.00234	−0.501170	+	0.865349i	$0.667097\pi$
$600$	0	0
$601$	5.98536e12	1.87135	0.935675	−	0.352863i	$-0.114792\pi$
$601$	5.98536e12	1.87135	0.935675	+	0.352863i	$0.114792\pi$
$602$	0	0
$603$	−1.02223e12	−0.314862
$604$	0	0
$605$	3.43436e12	1.04219
$606$	0	0
$607$	−1.60857e11	−0.0480939	−0.0240469	−	0.999711i	$-0.507655\pi$
$607$	−1.60857e11	−0.0480939	−0.0240469	+	0.999711i	$0.507655\pi$
$608$	0	0
$609$	1.92267e10	0.00566405
$610$	0	0
$611$	−2.41920e11	−0.0702243
$612$	0	0
$613$	−4.87119e12	−1.39336	−0.696679	−	0.717383i	$-0.745340\pi$
$613$	−4.87119e12	−1.39336	−0.696679	+	0.717383i	$0.745340\pi$
$614$	0	0
$615$	3.82152e12	1.07721
$616$	0	0
$617$	3.48094e12	0.966970	0.483485	−	0.875353i	$-0.339371\pi$
$617$	3.48094e12	0.966970	0.483485	+	0.875353i	$0.339371\pi$
$618$	0	0
$619$	−2.25847e12	−0.618310	−0.309155	−	0.951012i	$-0.600046\pi$
$619$	−2.25847e12	−0.618310	−0.309155	+	0.951012i	$0.600046\pi$
$620$	0	0
$621$	4.79935e12	1.29500
$622$	0	0
$623$	4.13285e11	0.109914
$624$	0	0
$625$	−4.13413e12	−1.08374
$626$	0	0
$627$	−1.04561e11	−0.0270189
$628$	0	0
$629$	3.13076e11	0.0797484
$630$	0	0
$631$	6.76730e12	1.69935	0.849676	−	0.527305i	$-0.176797\pi$
$631$	6.76730e12	1.69935	0.849676	+	0.527305i	$0.176797\pi$
$632$	0	0
$633$	−3.70709e12	−0.917734
$634$	0	0
$635$	2.02571e12	0.494419
$636$	0	0
$637$	−2.82960e12	−0.680923
$638$	0	0
$639$	1.82516e12	0.433060
$640$	0	0
$641$	7.28070e12	1.70338	0.851691	−	0.524045i	$-0.175578\pi$
$641$	7.28070e12	1.70338	0.851691	+	0.524045i	$0.175578\pi$
$642$	0	0
$643$	6.62565e12	1.52855	0.764274	−	0.644892i	$-0.223098\pi$
$643$	6.62565e12	1.52855	0.764274	+	0.644892i	$0.223098\pi$
$644$	0	0
$645$	−2.70261e12	−0.614843
$646$	0	0
$647$	6.36313e12	1.42758	0.713792	−	0.700358i	$-0.246976\pi$
$647$	6.36313e12	1.42758	0.713792	+	0.700358i	$0.246976\pi$
$648$	0	0
$649$	−3.52234e11	−0.0779346
$650$	0	0
$651$	1.95903e10	0.00427491
$652$	0	0
$653$	5.95209e12	1.28103	0.640516	−	0.767945i	$-0.278720\pi$
$653$	5.95209e12	1.28103	0.640516	+	0.767945i	$0.278720\pi$
$654$	0	0
$655$	8.58609e12	1.82268
$656$	0	0
$657$	−1.78260e12	−0.373259
$658$	0	0
$659$	5.45445e12	1.12659	0.563296	−	0.826255i	$-0.309533\pi$
$659$	5.45445e12	1.12659	0.563296	+	0.826255i	$0.309533\pi$
$660$	0	0
$661$	−2.82523e12	−0.575634	−0.287817	−	0.957685i	$-0.592930\pi$
$661$	−2.82523e12	−0.575634	−0.287817	+	0.957685i	$0.592930\pi$
$662$	0	0
$663$	−5.06008e11	−0.101706
$664$	0	0
$665$	−3.09005e11	−0.0612728
$666$	0	0
$667$	−8.74951e11	−0.171166
$668$	0	0
$669$	3.27643e12	0.632388
$670$	0	0
$671$	−3.12579e11	−0.0595263
$672$	0	0
$673$	−2.66233e12	−0.500259	−0.250129	−	0.968212i	$-0.580473\pi$
$673$	−2.66233e12	−0.500259	−0.250129	+	0.968212i	$0.580473\pi$
$674$	0	0
$675$	−4.95280e11	−0.0918298
$676$	0	0
$677$	−6.37169e12	−1.16575	−0.582875	−	0.812562i	$-0.698072\pi$
$677$	−6.37169e12	−1.16575	−0.582875	+	0.812562i	$0.698072\pi$
$678$	0	0
$679$	6.44065e11	0.116283
$680$	0	0
$681$	2.39115e12	0.426035
$682$	0	0
$683$	−7.29274e12	−1.28232	−0.641161	−	0.767406i	$-0.721547\pi$
$683$	−7.29274e12	−1.28232	−0.641161	+	0.767406i	$0.721547\pi$
$684$	0	0
$685$	−3.29469e12	−0.571752
$686$	0	0
$687$	3.05000e12	0.522390
$688$	0	0
$689$	−2.02806e12	−0.342843
$690$	0	0
$691$	−8.91264e12	−1.48715	−0.743576	−	0.668652i	$-0.766872\pi$
$691$	−8.91264e12	−1.48715	−0.743576	+	0.668652i	$0.766872\pi$
$692$	0	0
$693$	1.40696e10	0.00231730
$694$	0	0
$695$	−3.01670e12	−0.490457
$696$	0	0
$697$	2.54177e12	0.407933
$698$	0	0
$699$	3.97782e12	0.630229
$700$	0	0
$701$	3.09782e12	0.484534	0.242267	−	0.970210i	$-0.422109\pi$
$701$	3.09782e12	0.484534	0.242267	+	0.970210i	$0.422109\pi$
$702$	0	0
$703$	−1.77719e12	−0.274432
$704$	0	0
$705$	4.31099e11	0.0657244
$706$	0	0
$707$	7.74197e11	0.116537
$708$	0	0
$709$	5.42374e12	0.806104	0.403052	−	0.915177i	$-0.367949\pi$
$709$	5.42374e12	0.806104	0.403052	+	0.915177i	$0.367949\pi$
$710$	0	0
$711$	−5.18906e12	−0.761509
$712$	0	0
$713$	−8.91499e11	−0.129187
$714$	0	0
$715$	2.64020e11	0.0377799
$716$	0	0
$717$	−7.16833e12	−1.01294
$718$	0	0
$719$	5.08993e12	0.710284	0.355142	−	0.934812i	$-0.384433\pi$
$719$	5.08993e12	0.710284	0.355142	+	0.934812i	$0.384433\pi$
$720$	0	0
$721$	5.16720e11	0.0712109
$722$	0	0
$723$	−6.28829e11	−0.0855875
$724$	0	0
$725$	9.02925e10	0.0121375
$726$	0	0
$727$	−8.07173e12	−1.07167	−0.535836	−	0.844322i	$-0.680003\pi$
$727$	−8.07173e12	−1.07167	−0.535836	+	0.844322i	$0.680003\pi$
$728$	0	0
$729$	4.58327e12	0.601037
$730$	0	0
$731$	−1.79756e12	−0.232838
$732$	0	0
$733$	6.29376e12	0.805272	0.402636	−	0.915360i	$-0.368094\pi$
$733$	6.29376e12	0.805272	0.402636	+	0.915360i	$0.368094\pi$
$734$	0	0
$735$	5.04232e12	0.637290
$736$	0	0
$737$	−2.13339e11	−0.0266358
$738$	0	0
$739$	9.33473e11	0.115134	0.0575668	−	0.998342i	$-0.481666\pi$
$739$	9.33473e11	0.115134	0.0575668	+	0.998342i	$0.481666\pi$
$740$	0	0
$741$	2.87238e12	0.349993
$742$	0	0
$743$	1.42665e13	1.71739	0.858694	−	0.512489i	$-0.171276\pi$
$743$	1.42665e13	1.71739	0.858694	+	0.512489i	$0.171276\pi$
$744$	0	0
$745$	4.38358e12	0.521346
$746$	0	0
$747$	−6.58905e11	−0.0774248
$748$	0	0
$749$	−2.76113e11	−0.0320567
$750$	0	0
$751$	−1.42997e13	−1.64039	−0.820197	−	0.572081i	$-0.806136\pi$
$751$	−1.42997e13	−1.64039	−0.820197	+	0.572081i	$0.806136\pi$
$752$	0	0
$753$	−8.17206e12	−0.926305
$754$	0	0
$755$	−1.54092e13	−1.72592
$756$	0	0
$757$	−6.04819e12	−0.669413	−0.334706	−	0.942322i	$-0.608637\pi$
$757$	−6.04819e12	−0.669413	−0.334706	+	0.942322i	$0.608637\pi$
$758$	0	0
$759$	3.85037e11	0.0421128
$760$	0	0
$761$	5.22191e12	0.564414	0.282207	−	0.959353i	$-0.408933\pi$
$761$	5.22191e12	0.564414	0.282207	+	0.959353i	$0.408933\pi$
$762$	0	0
$763$	−8.10294e11	−0.0865530
$764$	0	0
$765$	−1.49941e12	−0.158287
$766$	0	0
$767$	9.67612e12	1.00954
$768$	0	0
$769$	−8.85529e12	−0.913134	−0.456567	−	0.889689i	$-0.650921\pi$
$769$	−8.85529e12	−0.913134	−0.456567	+	0.889689i	$0.650921\pi$
$770$	0	0
$771$	−6.19575e12	−0.631465
$772$	0	0
$773$	5.21750e12	0.525599	0.262799	−	0.964850i	$-0.415354\pi$
$773$	5.21750e12	0.525599	0.262799	+	0.964850i	$0.415354\pi$
$774$	0	0
$775$	9.20002e10	0.00916075
$776$	0	0
$777$	−1.43809e11	−0.0141544
$778$	0	0
$779$	−1.44285e13	−1.40379
$780$	0	0
$781$	3.80911e11	0.0366348
$782$	0	0
$783$	−1.37767e12	−0.130984
$784$	0	0
$785$	−1.24397e13	−1.16922
$786$	0	0
$787$	1.22879e13	1.14181	0.570903	−	0.821017i	$-0.306593\pi$
$787$	1.22879e13	1.14181	0.570903	+	0.821017i	$0.306593\pi$
$788$	0	0
$789$	1.47732e12	0.135715
$790$	0	0
$791$	−1.29054e12	−0.117213
$792$	0	0
$793$	8.58678e12	0.771082
$794$	0	0
$795$	3.61399e12	0.320874
$796$	0	0
$797$	1.04691e13	0.919063	0.459532	−	0.888161i	$-0.348017\pi$
$797$	1.04691e13	0.919063	0.459532	+	0.888161i	$0.348017\pi$
$798$	0	0
$799$	2.86733e11	0.0248895
$800$	0	0
$801$	−1.13838e13	−0.977107
$802$	0	0
$803$	−3.72029e11	−0.0315760
$804$	0	0
$805$	1.13788e12	0.0955024
$806$	0	0
$807$	9.50398e11	0.0788814
$808$	0	0
$809$	−1.43675e13	−1.17927	−0.589634	−	0.807671i	$-0.700728\pi$
$809$	−1.43675e13	−1.17927	−0.589634	+	0.807671i	$0.700728\pi$
$810$	0	0
$811$	1.55011e12	0.125825	0.0629127	−	0.998019i	$-0.479961\pi$
$811$	1.55011e12	0.125825	0.0629127	+	0.998019i	$0.479961\pi$
$812$	0	0
$813$	1.21347e13	0.974137
$814$	0	0
$815$	−1.00849e13	−0.800690
$816$	0	0
$817$	1.02039e13	0.801249
$818$	0	0
$819$	−3.86501e11	−0.0300174
$820$	0	0
$821$	−1.37949e13	−1.05968	−0.529839	−	0.848098i	$-0.677748\pi$
$821$	−1.37949e13	−1.05968	−0.529839	+	0.848098i	$0.677748\pi$
$822$	0	0
$823$	1.07251e13	0.814899	0.407449	−	0.913228i	$-0.366418\pi$
$823$	1.07251e13	0.814899	0.407449	+	0.913228i	$0.366418\pi$
$824$	0	0
$825$	−3.97347e10	−0.00298626
$826$	0	0
$827$	−1.65480e13	−1.23019	−0.615093	−	0.788454i	$-0.710882\pi$
$827$	−1.65480e13	−1.23019	−0.615093	+	0.788454i	$0.710882\pi$
$828$	0	0
$829$	1.03812e13	0.763398	0.381699	−	0.924287i	$-0.375339\pi$
$829$	1.03812e13	0.763398	0.381699	+	0.924287i	$0.375339\pi$
$830$	0	0
$831$	−9.69040e12	−0.704916
$832$	0	0
$833$	3.35374e12	0.241339
$834$	0	0
$835$	−1.19488e13	−0.850621
$836$	0	0
$837$	−1.40373e12	−0.0988595
$838$	0	0
$839$	6.25038e12	0.435489	0.217745	−	0.976006i	$-0.430130\pi$
$839$	6.25038e12	0.435489	0.217745	+	0.976006i	$0.430130\pi$
$840$	0	0
$841$	−1.42560e13	−0.982687
$842$	0	0
$843$	4.18428e12	0.285362
$844$	0	0
$845$	8.23589e12	0.555719
$846$	0	0
$847$	−1.04925e12	−0.0700496
$848$	0	0
$849$	1.04207e13	0.688356
$850$	0	0
$851$	6.54432e12	0.427742
$852$	0	0
$853$	−2.08264e13	−1.34693	−0.673464	−	0.739220i	$-0.735194\pi$
$853$	−2.08264e13	−1.34693	−0.673464	+	0.739220i	$0.735194\pi$
$854$	0	0
$855$	8.51147e12	0.544700
$856$	0	0
$857$	2.34103e12	0.148249	0.0741246	−	0.997249i	$-0.476384\pi$
$857$	2.34103e12	0.148249	0.0741246	+	0.997249i	$0.476384\pi$
$858$	0	0
$859$	1.14439e13	0.717139	0.358569	−	0.933503i	$-0.383265\pi$
$859$	1.14439e13	0.717139	0.358569	+	0.933503i	$0.383265\pi$
$860$	0	0
$861$	−1.16754e12	−0.0724031
$862$	0	0
$863$	1.65044e13	1.01286	0.506431	−	0.862281i	$-0.330965\pi$
$863$	1.65044e13	1.01286	0.506431	+	0.862281i	$0.330965\pi$
$864$	0	0
$865$	−2.87029e12	−0.174323
$866$	0	0
$867$	5.99739e11	0.0360476
$868$	0	0
$869$	−1.08295e12	−0.0644201
$870$	0	0
$871$	5.86056e12	0.345031
$872$	0	0
$873$	−1.77406e13	−1.03372
$874$	0	0
$875$	1.15554e12	0.0666419
$876$	0	0
$877$	−1.42710e11	−0.00814620	−0.00407310	−	0.999992i	$-0.501297\pi$
$877$	−1.42710e11	−0.00814620	−0.00407310	+	0.999992i	$0.501297\pi$
$878$	0	0
$879$	8.29777e12	0.468826
$880$	0	0
$881$	2.11785e13	1.18441	0.592206	−	0.805786i	$-0.298257\pi$
$881$	2.11785e13	1.18441	0.592206	+	0.805786i	$0.298257\pi$
$882$	0	0
$883$	−1.83970e13	−1.01841	−0.509207	−	0.860644i	$-0.670061\pi$
$883$	−1.83970e13	−1.01841	−0.509207	+	0.860644i	$0.670061\pi$
$884$	0	0
$885$	−1.72427e13	−0.944847
$886$	0	0
$887$	1.18951e13	0.645225	0.322613	−	0.946531i	$-0.395439\pi$
$887$	1.18951e13	0.645225	0.322613	+	0.946531i	$0.395439\pi$
$888$	0	0
$889$	−6.18888e11	−0.0332318
$890$	0	0
$891$	−1.43329e10	−0.000761874 0
$892$	0	0
$893$	−1.62765e12	−0.0856505
$894$	0	0
$895$	−3.66727e13	−1.91047
$896$	0	0
$897$	−1.05772e13	−0.545514
$898$	0	0
$899$	2.55908e11	0.0130667
$900$	0	0
$901$	2.40373e12	0.121513
$902$	0	0
$903$	8.25692e11	0.0413260
$904$	0	0
$905$	4.43377e12	0.219713
$906$	0	0
$907$	−3.91571e13	−1.92122	−0.960612	−	0.277893i	$-0.910364\pi$
$907$	−3.91571e13	−1.92122	−0.960612	+	0.277893i	$0.910364\pi$
$908$	0	0
$909$	−2.13251e13	−1.03599
$910$	0	0
$911$	2.83393e13	1.36319	0.681595	−	0.731730i	$-0.261287\pi$
$911$	2.83393e13	1.36319	0.681595	+	0.731730i	$0.261287\pi$
$912$	0	0
$913$	−1.37513e11	−0.00654977
$914$	0	0
$915$	−1.53015e13	−0.721672
$916$	0	0
$917$	−2.62319e12	−0.122509
$918$	0	0
$919$	1.73995e13	0.804668	0.402334	−	0.915493i	$-0.368199\pi$
$919$	1.73995e13	0.804668	0.402334	+	0.915493i	$0.368199\pi$
$920$	0	0
$921$	1.13119e13	0.518044
$922$	0	0
$923$	−1.04639e13	−0.474554
$924$	0	0
$925$	−6.75356e11	−0.0303316
$926$	0	0
$927$	−1.42329e13	−0.633047
$928$	0	0
$929$	−3.95509e13	−1.74215	−0.871074	−	0.491151i	$-0.836576\pi$
$929$	−3.95509e13	−1.74215	−0.871074	+	0.491151i	$0.836576\pi$
$930$	0	0
$931$	−1.90377e13	−0.830501
$932$	0	0
$933$	−2.74988e11	−0.0118808
$934$	0	0
$935$	−3.12926e11	−0.0133903
$936$	0	0
$937$	4.31185e13	1.82741	0.913704	−	0.406381i	$-0.133210\pi$
$937$	4.31185e13	1.82741	0.913704	+	0.406381i	$0.133210\pi$
$938$	0	0
$939$	−1.37504e13	−0.577194
$940$	0	0
$941$	1.23028e13	0.511505	0.255753	−	0.966742i	$-0.417677\pi$
$941$	1.23028e13	0.511505	0.255753	+	0.966742i	$0.417677\pi$
$942$	0	0
$943$	5.31313e13	2.18800
$944$	0	0
$945$	1.79167e12	0.0730828
$946$	0	0
$947$	−3.99251e13	−1.61314	−0.806568	−	0.591142i	$-0.798677\pi$
$947$	−3.99251e13	−1.61314	−0.806568	+	0.591142i	$0.798677\pi$
$948$	0	0
$949$	1.02199e13	0.409024
$950$	0	0
$951$	5.82424e12	0.230902
$952$	0	0
$953$	−4.12885e13	−1.62148	−0.810739	−	0.585408i	$-0.800934\pi$
$953$	−4.12885e13	−1.62148	−0.810739	+	0.585408i	$0.800934\pi$
$954$	0	0
$955$	−4.60930e13	−1.79316
$956$	0	0
$957$	−1.10526e11	−0.00425953
$958$	0	0
$959$	1.00658e12	0.0384296
$960$	0	0
$961$	−2.61789e13	−0.990138
$962$	0	0
$963$	7.60548e12	0.284976
$964$	0	0
$965$	−3.35241e12	−0.124447
$966$	0	0
$967$	3.74952e13	1.37898	0.689488	−	0.724297i	$-0.257836\pi$
$967$	3.74952e13	1.37898	0.689488	+	0.724297i	$0.257836\pi$
$968$	0	0
$969$	−3.40444e12	−0.124048
$970$	0	0
$971$	1.85734e13	0.670508	0.335254	−	0.942128i	$-0.391178\pi$
$971$	1.85734e13	0.670508	0.335254	+	0.942128i	$0.391178\pi$
$972$	0	0
$973$	9.21653e11	0.0329655
$974$	0	0
$975$	1.09154e12	0.0386829
$976$	0	0
$977$	−5.09643e13	−1.78954	−0.894769	−	0.446530i	$-0.852660\pi$
$977$	−5.09643e13	−1.78954	−0.894769	+	0.446530i	$0.852660\pi$
$978$	0	0
$979$	−2.37580e12	−0.0826586
$980$	0	0
$981$	2.23194e13	0.769434
$982$	0	0
$983$	−1.27845e12	−0.0436710	−0.0218355	−	0.999762i	$-0.506951\pi$
$983$	−1.27845e12	−0.0436710	−0.0218355	+	0.999762i	$0.506951\pi$
$984$	0	0
$985$	−3.87578e13	−1.31189
$986$	0	0
$987$	−1.31708e11	−0.00441759
$988$	0	0
$989$	−3.75748e13	−1.24886
$990$	0	0
$991$	−1.53850e13	−0.506719	−0.253360	−	0.967372i	$-0.581536\pi$
$991$	−1.53850e13	−0.506719	−0.253360	+	0.967372i	$0.581536\pi$
$992$	0	0
$993$	−1.06514e13	−0.347643
$994$	0	0
$995$	−2.12920e13	−0.688672
$996$	0	0
$997$	3.56093e13	1.14139	0.570696	−	0.821162i	$-0.306674\pi$
$997$	3.56093e13	1.14139	0.570696	+	0.821162i	$0.306674\pi$
$998$	0	0
$999$	1.03045e13	0.327327

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	272.10.a.f.1.4		5
4.3	odd	2		17.10.a.a.1.5	✓	5
12.11	even	2		153.10.a.c.1.1		5
68.67	odd	2		289.10.a.a.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.a.a.1.5	✓	5	4.3	odd	2
153.10.a.c.1.1		5	12.11	even	2
272.10.a.f.1.4		5	1.1	even	1	trivial
289.10.a.a.1.5		5	68.67	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 272 = 2^{4} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	272.a (trivial)

\(f(q)\)	\(=\)	\(q+85.9747 q^{3} -1460.58 q^{5} +446.232 q^{7} -12291.3 q^{9} +O(q^{10})\)	\(q+85.9747 q^{3} -1460.58 q^{5} +446.232 q^{7} -12291.3 q^{9} -2565.20 q^{11} +70467.9 q^{13} -125573. q^{15} -83521.0 q^{17} +474111. q^{19} +38364.6 q^{21} -1.74586e6 q^{23} +180168. q^{25} -2.74899e6 q^{27} +501157. q^{29} +510635. q^{31} -220542. q^{33} -651757. q^{35} -3.74847e6 q^{37} +6.05845e6 q^{39} -3.04327e7 q^{41} +2.15222e7 q^{43} +1.79525e7 q^{45} -3.43306e6 q^{47} -4.01545e7 q^{49} -7.18069e6 q^{51} -2.87800e7 q^{53} +3.74668e6 q^{55} +4.07615e7 q^{57} +1.37313e8 q^{59} +1.21854e8 q^{61} -5.48479e6 q^{63} -1.02924e8 q^{65} +8.31665e7 q^{67} -1.50100e8 q^{69} -1.48492e8 q^{71} +1.45029e8 q^{73} +1.54899e7 q^{75} -1.14467e6 q^{77} +4.22172e8 q^{79} +5.58742e6 q^{81} +5.36072e7 q^{83} +1.21989e8 q^{85} +4.30868e7 q^{87} +9.26167e8 q^{89} +3.14450e7 q^{91} +4.39017e7 q^{93} -6.92477e8 q^{95} +1.44334e9 q^{97} +3.15298e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 236 q^{3} + 1480 q^{5} + 13202 q^{7} + 10981 q^{9}+O(q^{10}) \) 5 * q + 236 * q^3 + 1480 * q^5 + 13202 * q^7 + 10981 * q^9	\( 5 q + 236 q^{3} + 1480 q^{5} + 13202 q^{7} + 10981 q^{9} + 68036 q^{11} - 158862 q^{13} + 687324 q^{15} - 417605 q^{17} + 370992 q^{19} + 1783880 q^{21} - 1645870 q^{23} + 3270239 q^{25} + 2998268 q^{27} + 3668616 q^{29} + 7262362 q^{31} - 11334900 q^{33} + 26503988 q^{35} - 31420708 q^{37} + 42449884 q^{39} - 7996938 q^{41} + 56908268 q^{43} + 12799536 q^{45} + 16903336 q^{47} - 11784059 q^{49} - 19710956 q^{51} - 83362982 q^{53} - 6363364 q^{55} + 136615904 q^{57} + 37946604 q^{59} - 77685452 q^{61} + 191945278 q^{63} - 40321288 q^{65} + 304503600 q^{67} - 333409272 q^{69} + 476602922 q^{71} - 289980486 q^{73} + 153685772 q^{75} - 143385648 q^{77} + 828240610 q^{79} + 891328609 q^{81} - 194681148 q^{83} - 123611080 q^{85} - 158149884 q^{87} + 376848106 q^{89} - 194543664 q^{91} + 3494835920 q^{93} - 1498679864 q^{95} + 692035246 q^{97} - 2027106408 q^{99}+O(q^{100}) \) 5 * q + 236 * q^3 + 1480 * q^5 + 13202 * q^7 + 10981 * q^9 + 68036 * q^11 - 158862 * q^13 + 687324 * q^15 - 417605 * q^17 + 370992 * q^19 + 1783880 * q^21 - 1645870 * q^23 + 3270239 * q^25 + 2998268 * q^27 + 3668616 * q^29 + 7262362 * q^31 - 11334900 * q^33 + 26503988 * q^35 - 31420708 * q^37 + 42449884 * q^39 - 7996938 * q^41 + 56908268 * q^43 + 12799536 * q^45 + 16903336 * q^47 - 11784059 * q^49 - 19710956 * q^51 - 83362982 * q^53 - 6363364 * q^55 + 136615904 * q^57 + 37946604 * q^59 - 77685452 * q^61 + 191945278 * q^63 - 40321288 * q^65 + 304503600 * q^67 - 333409272 * q^69 + 476602922 * q^71 - 289980486 * q^73 + 153685772 * q^75 - 143385648 * q^77 + 828240610 * q^79 + 891328609 * q^81 - 194681148 * q^83 - 123611080 * q^85 - 158149884 * q^87 + 376848106 * q^89 - 194543664 * q^91 + 3494835920 * q^93 - 1498679864 * q^95 + 692035246 * q^97 - 2027106408 * q^99

Introduction

L-functions

Modular forms

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Database

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